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    <meta name="description" content="在上一篇文章的最后，我们留了一个问题，就是如果仅保留等腰的条件，是否还有比较好的结论？ 要解决这个问题，我们先从一种特殊情况谈起．">
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    <div class="post-body" itemprop="articleBody"><p>在上一篇文章的最后，我们留了一个问题，就是如果仅保留等腰的条件，是否还有比较好的结论？</p>
<p>要解决这个问题，我们先从一种特殊情况谈起．</p>
<span id="more"></span>
<h2 id="1-一种特殊情况"><a class="header-anchor" href="#1-一种特殊情况"></a>1. 一种特殊情况</h2>
<p>当两个等腰 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>B</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">\mathrm{Rt} \triangle ABD</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathrm">Rt</span></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\triangle ACF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span> 旋转的时候，一种非常特殊的情况就是两个三角形的斜边共线的情况，如<code>图1</code>：</p>
<p><img data-src="image-20200403021215357.png" alt="图1" /></p>
<p>这个时候上面的结论依旧成立，而且我们注意到这个时候 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>B</mi><mi>A</mi><mi>C</mi><mo>=</mo><mn>9</mn><msup><mn>0</mn><mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\angle BAC = 90^\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span>，因此如果我们取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">BC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span> 的中点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span>，则有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mi>B</mi><mi>C</mi><mo>=</mo><mi>Q</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">QA = \dfrac{1}{2} BC = QP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span></span></span></span>，也就是说 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>P</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\triangle APQ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal">PQ</span></span></span></span> 是一个等腰三角形．</p>
<p>熟悉四点共圆的朋友马上就会想到，这里面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">ABCP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">BCP</span></span></span></span> 四点共圆，圆心恰好就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span>．</p>
<p><img data-src="image-20200403021414962.png" alt="图2" /></p>
<h3 id="1-1-逆命题"><a class="header-anchor" href="#1-1-逆命题"></a>1.1. 逆命题</h3>
<p>那我们反过来想一下，如果取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">\triangle ABC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span> 的外接圆和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 交于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 点是否一定是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点？</p>
<p><img data-src="image-20200403124126297.png" alt="图3" /></p>
<p>如果考虑同一法的话，很明显这个结论是成立的．</p>
<p>那如果不用同一法呢？</p>
<h3 id="1-2-构造梯形的中位线"><a class="header-anchor" href="#1-2-构造梯形的中位线"></a>1.2. 构造梯形的中位线</h3>
<p>一种方法是构造梯形的中位线．我们分别过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> 作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的垂线，垂足依次为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">CT</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">CT</span></span></span></span> 是梯形 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>M</mi><mi>N</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">BMNC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">BMNC</span></span></span></span> 的中位线，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> 分别是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">AD</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">AF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">AP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 的中点，于是</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mi>D</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mi>M</mi><mi>T</mi><mo>=</mo><mn>2</mn><mi>N</mi><mi>T</mi><mo>=</mo><mn>2</mn><mrow><mo fence="true">(</mo><mi>A</mi><mi>N</mi><mo>−</mo><mi>A</mi><mi>T</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>A</mi><mi>F</mi><mo>−</mo><mi>A</mi><mi>P</mi><mo>=</mo><mi>P</mi><mi>F</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
  PD &amp;= 2MT = 2NT = 2 \left( AN - AT \right) \\
  &amp;= AF - AP = PF
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.13889em;">MT</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.13889em;">NT</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">PF</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点．</p>
<p><img data-src="image-20200403124523191.png" alt="图4" /></p>
<p>另一种方法是构造旋转相似，这种方法放到我们后面的推广里来讲．</p>
<h2 id="2-一种推广"><a class="header-anchor" href="#2-一种推广"></a>2. 一种推广</h2>
<p>如果要保持四点共圆的条件不变，我们可以把条件弱化成什么样子？</p>
<p>注意如果要保持四点共圆的话，我们要保持 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>B</mi><mi>A</mi><mi>C</mi><mo>=</mo><mn>9</mn><msup><mn>0</mn><mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\angle BAC = 90^\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span>，因此两个等腰三角形的两底角要保持互余，也就是两顶角要保持互补．</p>
<p>因此，我们可以把两个等腰直角三角形的条件改为，两个「顶角互补的等腰三角形」：</p>
<p><img data-src="image-20200404001633874.png" alt="图5" /></p>
<p>在<code>图5</code>中，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mi>B</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">AB = BD</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>C</mi><mo>=</mo><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">AC = CF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>A</mi><mi>B</mi><mi>D</mi><mo>+</mo><mi mathvariant="normal">∠</mi><mi>A</mi><mi>C</mi><mi>F</mi><mo>=</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\angle ABD + \angle ACF = 180^\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7756em;vertical-align:-0.0833em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span>，我们分别取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">BC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>B</mi><mi>P</mi><mi>C</mi><mo>=</mo><mn>9</mn><msup><mn>0</mn><mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\angle BPC = 90^\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.07153em;">BPC</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span>，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>Q</mi><mo>=</mo><mi>Q</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">AQ = QP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span></span></span></span>．</p>
<p>我们可以利用上一篇文章中的三种方法，对这种情况进行证明．因为方法几乎是一样的，这里就从略了．</p>
<h2 id="3-继续推广"><a class="header-anchor" href="#3-继续推广"></a>3. 继续推广</h2>
<p>我们回顾一下<code>图4</code>的证明，这种方法本质上就是用了三个等腰 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>B</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">\triangle ABD</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\triangle ACF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>P</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\triangle APQ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal">PQ</span></span></span></span> 的条件，因此我们可以把条件再进行弱化，如下图：</p>
<p><img data-src="image-20200403125819232.png" alt="图6" /></p>
<p>如<code>图6</code>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>B</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">\triangle ABD</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\triangle ACF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span> 都是等腰三角形，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mi>B</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">AB = BD</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>C</mi><mo>=</mo><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">AC = CF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">BC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span> 的中点，则</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mi>Q</mi><mi>P</mi><mo>⟺</mo><mi>P</mi><mi>D</mi><mo>=</mo><mi>P</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">QA = QP \Longleftrightarrow PD = PF
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">PF</span></span></span></span></span></p>
<p>我们来想一下，前面的那些方法，是否还成立？</p>
<div class="note primary"><p>同时，大家可以想一下，这个图形和我们上一篇文章中里的图形有什么区别和联系．</p>
</div>
<h3 id="3-1-倍长中线"><a class="header-anchor" href="#3-1-倍长中线"></a>3.1. 倍长中线</h3>
<p>这个时候，倍长中线的方法依旧可以使用，不过这个时候，应该要倍长 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">AQ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">Q</span></span></span></span>．</p>
<p><img data-src="image-20200404002901935.png" alt="图7" /></p>
<blockquote>
<p>为了两个全等三角形看着大一些，我调整了两个等腰三角形的角度，但这不影响我们的证明</p>
</blockquote>
<p>如<code>图7</code>，倍长 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">AQ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">Q</span></span></span></span> 至 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span>，可以证明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>D</mi><mi>B</mi><mi>K</mi><mo>≅</mo><mi mathvariant="normal">△</mi><mi>K</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\triangle DBK \cong \triangle KCF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≅</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span>．</p>
<p>证明这个全等的关键，是要证明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>D</mi><mi>B</mi><mi>K</mi><mo>=</mo><mi mathvariant="normal">∠</mi><mi>K</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\angle DBK = \angle KCF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span>．为了叙述方便，我们设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>D</mi><mi>B</mi><mi>A</mi><mo>=</mo><mn>2</mn><mi>α</mi></mrow><annotation encoding="application/x-tex">\angle DBA = 2\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>A</mi><mi>C</mi><mi>F</mi><mo>=</mo><mn>2</mn><mi>β</mi></mrow><annotation encoding="application/x-tex">\angle ACF = 2\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>B</mi><mi>A</mi><mi>C</mi><mo>=</mo><mi>α</mi><mo>+</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\angle BAC = \alpha + \beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>A</mi><mi>B</mi><mi>K</mi><mo>=</mo><mi mathvariant="normal">∠</mi><mi>A</mi><mi>C</mi><mi>K</mi><mo>=</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>−</mo><mi>α</mi><mo>−</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\angle ABK = \angle ACK = 180^\circ - \alpha - \beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7574em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span>，于是</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∠</mi><mi>D</mi><mi>B</mi><mi>K</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">∠</mi><mi>D</mi><mi>B</mi><mi>A</mi><mo>+</mo><mi mathvariant="normal">∠</mi><mi>A</mi><mi>B</mi><mi>K</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mi>α</mi><mo>+</mo><mo stretchy="false">(</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>−</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>+</mo><mi>α</mi><mo>−</mo><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∠</mi><mi>K</mi><mi>C</mi><mi>F</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>36</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>−</mo><mi mathvariant="normal">∠</mi><mi>A</mi><mi>C</mi><mi>F</mi><mo>−</mo><mi mathvariant="normal">∠</mi><mi>A</mi><mi>C</mi><mi>K</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>36</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>−</mo><mn>2</mn><mi>β</mi><mo>−</mo><mo stretchy="false">(</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>−</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>+</mo><mi>α</mi><mo>−</mo><mi>β</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
  \angle DBK &amp;= \angle DBA + \angle ABK \\
  &amp;= 2\alpha + (180^\circ - \alpha - \beta) \\
  &amp;= 180^\circ + \alpha - \beta \\
  \\
  \angle KCF &amp;= 360^\circ - \angle ACF - \angle ACK \\
  &amp;= 360^\circ - 2\beta - (180^\circ - \alpha - \beta) \\
  &amp;= 180^\circ + \alpha - \beta
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.5em;vertical-align:-5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5em;"><span style="top:-7.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:1.34em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5em;"><span style="top:-7.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">36</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">36</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span></span></span><span style="top:1.34em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>故 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>D</mi><mi>B</mi><mi>K</mi><mo>=</mo><mi mathvariant="normal">∠</mi><mi>K</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\angle DBK = \angle KCF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">CF</span></span></span></span>．</p>
<p>有了全等之后，我们就有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>D</mi><mo>=</mo><mi>K</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">KD = KF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">KD</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>．</p>
<p>如果已知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点，由「三线合一」可知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>P</mi><mo>⊥</mo><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">KP \perp DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>K</mi><mi>P</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\triangle KPA</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal">A</span></span></span></span> 是直角三角形，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">KQ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal">Q</span></span></span></span> 是其斜边中线，于是</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>A</mi><mi>K</mi><mo>=</mo><mi>A</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">PQ = \dfrac{1}{2} AK = AQ
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">PQ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">Q</span></span></span></span></span></p>
<p>如果已知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mi>Q</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">QA  = QP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span></span></span></span>，由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mi>K</mi><mo>=</mo><mi>Q</mi><mi>A</mi><mo>=</mo><mi>Q</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">QK = QA = QP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span></span></span></span> 可知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>K</mi><mi>P</mi><mi>A</mi><mo>=</mo><mn>9</mn><msup><mn>0</mn><mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\angle KPA = 90^\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span></span></span></span>，再由「三线合一」可知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>D</mi><mo>=</mo><mi>P</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">PD = PF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">PF</span></span></span></span>．</p>
<h3 id="3-2-构造三角形的中位线"><a class="header-anchor" href="#3-2-构造三角形的中位线"></a>3.2. 构造三角形的中位线</h3>
<p>这个时候我们没有办法利用中位线直接证明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mi>Q</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">QA = QP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span></span></span></span>，但是我们可以分别取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">AD</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">AF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>，先证明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mi>M</mi><mo>=</mo><mi>Q</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">QM= QN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">QM</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">QN</span></span></span></span>．</p>
<p><img data-src="image-20200404005413523.png" alt="图8" /></p>
<p>我们分别取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">AC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> 的中点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>，于是可以证明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>M</mi><mi>S</mi><mi>Q</mi><mo>≅</mo><mi mathvariant="normal">△</mi><mi>Q</mi><mi>T</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">\triangle MSQ \cong \triangle QTN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">MSQ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≅</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.10903em;">QTN</span></span></span></span>．其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>M</mi><mi>S</mi><mi>Q</mi><mo>=</mo><mi mathvariant="normal">∠</mi><mi>Q</mi><mi>T</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">\angle MSQ = \angle QTN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord">∠</span><span class="mord mathnormal">MSQ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.10903em;">QTN</span></span></span></span> 的证明和上一个方法类似．</p>
<p>于是接下来，只需要证明 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>Q</mi><mi>A</mi><mi>M</mi><mo>≅</mo><mi mathvariant="normal">△</mi><mi>Q</mi><mi>P</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">\triangle QAM \cong \triangle QPN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≅</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.10903em;">QPN</span></span></span></span> 即可（或者过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> 作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">AP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 的垂线，利用三线合一来做）．</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mi>D</mi><mo>=</mo><mi>P</mi><mi>F</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>A</mi><mi>M</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>A</mi><mi>D</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>D</mi><mi>F</mi><mo>−</mo><mi>A</mi><mi>F</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mi>F</mi><mo>−</mo><mi>N</mi><mi>F</mi><mo>=</mo><mi>P</mi><mi>N</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">S</mi></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mi>Q</mi><mi>P</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">S</mi></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
  PD = PF &amp;\implies AM = \frac{1}{2} AD \\
  &amp;= \frac{1}{2}(DF - AF) \\
  &amp;= PF - NF = PN \\
  &amp;\implies \mathrm{SAS} \\
  \\
  QA = QP &amp;\implies \mathrm{AAS}
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.6149em;vertical-align:-5.0574em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5574em;"><span style="top:-7.5574em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">PF</span></span></span><span style="top:-5.25em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-3.424em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-1.924em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-0.424em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:1.076em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.0574em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5574em;"><span style="top:-7.5574em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-5.25em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mclose">)</span></span></span><span style="top:-3.424em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">PF</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">NF</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">PN</span></span></span><span style="top:-1.924em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathrm">SAS</span></span></span></span><span style="top:1.076em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathrm">AAS</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.0574em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<h3 id="3-3-两种方法的联系"><a class="header-anchor" href="#3-3-两种方法的联系"></a>3.3. 两种方法的联系</h3>
<p>实际上，如果我们把上一种方法的图和这一种方法的图放在一起，就会发现这两组三角形其实是相似的．</p>
<p><img data-src="image-20200404010445766.png" alt="图9" /></p>
<h3 id="3-4-构造梯形的中位线"><a class="header-anchor" href="#3-4-构造梯形的中位线"></a>3.4. 构造梯形的中位线</h3>
<p>这种方法和<code>图4</code>的证明是一样的，这里就不在重复了．</p>
<p>可以看出，这是最简单的一种证明方法．</p>
<h3 id="3-5-构造旋转相似"><a class="header-anchor" href="#3-5-构造旋转相似"></a>3.5. 构造旋转相似</h3>
<p>这个图还有一个证明方法，就是构造 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">BC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span> 的对称点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>，如<code>图10</code>，则</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∠</mi><mi>S</mi><mi>B</mi><mi>D</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mi mathvariant="normal">∠</mi><mi>S</mi><mi>A</mi><mi>D</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mrow><mo fence="true">(</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>−</mo><mi mathvariant="normal">∠</mi><mi>S</mi><mi>A</mi><mi>F</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">∠</mi><mi>S</mi><mi>C</mi><mi>F</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
  \angle SBD &amp;= 2\angle SAD \\
  &amp;= 2\left( 180^\circ - \angle SAF \right) \\
  &amp;= \angle SCF
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5em;vertical-align:-2em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2∠</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.13889em;">SCF</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>于是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>D</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\triangle SBD \sim \triangle SCF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.13889em;">SCF</span></span></span></span>，这是一组旋转相似．</p>
<p><img data-src="image-20200403130658955.png" alt="图10" /></p>
<p>如果已知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点，那么就有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>D</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>Q</mi><mi>P</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>C</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\triangle SBD \sim \triangle SQP \sim \triangle SCF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.13889em;">SQP</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.13889em;">SCF</span></span></span></span>，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>Q</mi><mo>=</mo><mi>S</mi><mi>Q</mi><mo>=</mo><mi>P</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">AQ = SQ = PQ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">SQ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">PQ</span></span></span></span>．</p>
<div class="note info"><p>具体的推导过程如下：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>D</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>C</mi><mi>F</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>C</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>D</mi><mi>F</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>Q</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>D</mi><mi>P</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>D</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>Q</mi><mi>P</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
  &amp; \triangle SBD \sim \triangle SCF \\
  \implies &amp; \triangle SBC \sim \triangle SDF \\
  \implies &amp; \triangle SBQ \sim \triangle SDP \\
  \implies &amp; \triangle SBD \sim \triangle SQP
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6em;vertical-align:-2.75em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.13889em;">SCF</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.07153em;">SBC</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord">△</span><span class="mord mathnormal">SBQ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.13889em;">SQP</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
</div>
<p>如果已知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mi>A</mi><mo>=</mo><mi>Q</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">QA  = QP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">QP</span></span></span></span>，那么</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∠</mi><mi>S</mi><mi>B</mi><mi>D</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mi mathvariant="normal">∠</mi><mi>S</mi><mi>A</mi><mi>D</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mrow><mo fence="true">(</mo><mn>18</mn><msup><mn>0</mn><mo>∘</mo></msup><mo>−</mo><mi mathvariant="normal">∠</mi><mi>S</mi><mi>A</mi><mi>P</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">∠</mi><mi>S</mi><mi>Q</mi><mi>P</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
  \angle SBD &amp;= 2\angle SAD \\
  &amp;= 2\left( 180^\circ - \angle SAP \right) \\
  &amp;= \angle SQP
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5em;vertical-align:-2em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2∠</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∘</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.13889em;">SQP</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>于是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>D</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>P</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\triangle SBD \sim \triangle SPQ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">SPQ</span></span></span></span>．由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">BC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span> 中点，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点．</p>
<div class="note info"><p>具体的推导过程如下：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mrow><mtable rowspacing="0.16em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>D</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>C</mi><mi>F</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>C</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>D</mi><mi>F</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>D</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>Q</mi><mi>P</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi mathvariant="normal">△</mi><mi>S</mi><mi>B</mi><mi>Q</mi><mo>∼</mo><mi mathvariant="normal">△</mi><mi>S</mi><mi>D</mi><mi>P</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">}</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>  </mtext><mtext>  </mtext><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>D</mi><mi>F</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>S</mi><mi>B</mi></mrow><mrow><mi>S</mi><mi>D</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mi>Q</mi></mrow><mrow><mi>D</mi><mi>P</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
  &amp;\left.
  \begin{array}{r}
    \triangle SBD \sim \triangle SCF \\
    \implies \triangle SBC \sim \triangle SDF \\
    \\
    \triangle SBD \sim \triangle SQP \\
    \implies \triangle SBQ \sim \triangle SDP
  \end{array}
  \right\} \\
  \\
  &amp; \;\;\implies \frac{BC}{DF} = \frac{SB}{SD} = \frac{BQ}{DP}
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.1464em;vertical-align:-4.8232em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.3232em;"><span style="top:-7.3232em;"><span class="pstrut" style="height:5.25em;"></span><span class="mord"></span></span><span style="top:-3.4332em;"><span class="pstrut" style="height:5.25em;"></span><span class="mord"></span></span><span style="top:-1.4128em;"><span class="pstrut" style="height:5.25em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.8232em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.3232em;"><span style="top:-7.3232em;"><span class="pstrut" style="height:5.25em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.13889em;">SCF</span></span></span><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.07153em;">SBC</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.13889em;">SQP</span></span></span><span style="top:-0.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal">SBQ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">△</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-1.366em;"><span class="pstrut" style="height:3.216em;"></span><span class="delimsizinginner delim-size4"><span>⎭</span></span></span><span style="top:-1.358em;"><span class="pstrut" style="height:3.216em;"></span><span style="height:1.216em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='1.216em' style='width:0.8889em' viewBox='0 0 888.89 1216' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V1216 H384z M384 0 H504 V1216 H384z'/></svg></span></span><span style="top:-3.216em;"><span class="pstrut" style="height:3.216em;"></span><span class="delimsizinginner delim-size4"><span>⎬</span></span></span><span style="top:-4.358em;"><span class="pstrut" style="height:3.216em;"></span><span style="height:1.216em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='1.216em' style='width:0.8889em' viewBox='0 0 888.89 1216' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V1216 H384z M384 0 H504 V1216 H384z'/></svg></span></span><span style="top:-5.566em;"><span class="pstrut" style="height:3.216em;"></span><span class="delimsizinginner delim-size4"><span>⎫</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-1.4128em;"><span class="pstrut" style="height:5.25em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">SB</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">BQ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.8232em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>因此</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>D</mi><mi>P</mi></mrow><mrow><mi>D</mi><mi>F</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mi>Q</mi></mrow><mrow><mi>B</mi><mi>P</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{DP}{DF} = \frac{BQ}{BP} = \frac{1}{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">BP</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">BQ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">DF</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的中点．</p>
</div>
<p><img data-src="image-20200403132128065.png" alt="图11" /></p>
<p>这种方法大量用到了相似和圆周角的性质，由此又可以引申出关于双圆问题的一些结论．有兴趣的朋友可以自行探究一下．</p>
<h3 id="3-6-和前文图形的关系"><a class="header-anchor" href="#3-6-和前文图形的关系"></a>3.6. 和前文图形的关系</h3>
<p>我们回过头来看一下<code>图8</code>，注意在这个图中我们平没有用到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 这两个点，因此我们考虑把这两个点去掉，于是这个图就变成了下面这样：</p>
<p><img data-src="image-20200404013143185.png" alt="图12" /></p>
<p>如<code>图12</code>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>M</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">\triangle AMB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">MB</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi><mi>A</mi><mi>N</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">\triangle ANC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">NC</span></span></span></span> 都是直角三角形，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> 共线，取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">BC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span></span> 的中点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mi>M</mi><mo>=</mo><mi>Q</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">QM = QN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">QM</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">QN</span></span></span></span>．</p>
<p>对比一下我们前面那篇文章中的<code>图11</code>，那个图的条件实际上两个直角三角形加上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi><mi>M</mi><mi>A</mi><mi>B</mi><mo>=</mo><mi mathvariant="normal">∠</mi><mi>C</mi><mi>A</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">\angle MAB = \angle CAN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>，而<code>图12</code>中则加上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> 共线，两个条件不同，结论却是相同的．</p>
<hr />
<p>至此，这个模型的讨论就暂时告一段落．我们在这三篇文章中，系统地讨论了和两个正方形相关的中点问题，以及他们的推广和变形．我们看到，解决这类问题的关键点在于处理好「中点」的条件．我们需要熟练地掌握「倍长中线」和「中位线」的技巧，才能够顺利地解决这些问题．</p>

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